How can I roll a cube (around its edges) with armatures?
This is just what you need! This is a Wolfram Alpha widget that showe you the cylogons (roulletes on n-gons about a straight line). The square is a default option:

http://demonstrations.wolfram.com/TracingACyclogonRouletteOfAPolygonRollingAlongALine/

EDIT: I made a canned animation of this rolling motion here, maybe you can use it:

http://pasteall.org/blend/index.php?id=51926